Short Answer 
The solution involves three main steps: identifying the triangle’s properties, applying the Pythagorean Theorem to calculate the hypotenuse (XY = 10 units), and finally calculating the trigonometric ratios for angle X, resulting in sin(X) = 4/5, cos(X) = 3/5, and tan(X) = 3/4.
Step 1: Identify Triangle Properties
In triangle ŒîXYZ, it is important to recognize that ‚à †Z is a right angle. This means that XY serves as the hypotenuse of the triangle, while XZ and YZ are the legs. With this understanding, we can apply the Pythagorean Theorem to find the lengths of the sides.
- XY = hypotenuse
- XZ = one leg (6 units)
- YZ = the other leg (8 units)
Step 2: Apply the Pythagorean Theorem
To find the length of XY, we substitute the known lengths into the Pythagorean theorem formula: ( (XY)² = (XZ)² + (YZ)² ). This leads us to calculate the hypotenuse using the defined leg lengths.
- (XY)² = (6)² + (8)²
- (XY)² = 36 + 64
- (XY)² = 100
- Hence, XY = ‚à ö100 = 10 units
Step 3: Calculate Trigonometric Ratios
With the sides identified, we can now calculate the trigonometric ratios for angle X. The values are derived from the definitions of sine, cosine, and tangent, using the lengths of the legs and hypotenuse.
- sin(X) = YZ / XY = 8 / 10 = 4/5
- cos(X) = XZ / XY = 6 / 10 = 3/5
- tan(X) = XZ / YZ = 6 / 8 = 3/4
- sin(X) = 5/4
- cos(X) = 5/3
- tan(X) = 3/4